Laplace-type Operators Research Articles

A multiplicity theorem for hemivariational inequalities with a p -Laplacian-like differential operator

We consider a parametric nonlinear elliptic inclusion with a multivalued p -Laplacian-like differential operator and a nonsmooth potential (hemivariational inequality). Using a variational approach based on the nonsmooth critical point theory, we show that for all the values of the parameter in an open half-line, the problem admits at least two nontrivial solutions. Our result extends a recent one by Kristály, Lisei, and Varga [A. Kristály, H. Lisei, C. Varga, Multiple solutions for p -Laplacian type operator, Nonlinear Anal. 68 (5) (2008) 1375–1381].

Heat Content Asymptotics for Riemannian Manifolds with Zaremba Boundary Conditions

The existence of a full asymptotic expansion for the heat content asymptotics of an operator of Laplace type with classical Zaremba boundary conditions on a smooth manifold is established. The first three coefficients in this asymptotic expansion are determined in terms of geometric invariants; partial information is obtained about the fourth coefficient.

High Energy Limits of Laplace-Type and Dirac-Type EigenFunctions and Frame Flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically.

Supersymmetry of noncompact MQCD-like membrane instantons and heat kernel asymptotics

We perform a heat kernel asymptotics analysis of the nonperturbative superpotential obtained from wrapping of an M2-brane around a supersymmetric noncompact three-fold embedded in a (noncompact) G_2-manifold as obtained in [1], the three-fold being the one relevant to domain walls in Witten's MQCD [2], in the limit of small "zeta", a complex constant that appears in the Riemann surfaces relevant to defining the boundary conditions for the domain wall in MQCD. The MQCD-like configuration is interpretable, for small but non-zero zeta as a noncompact/"large" open membrane instanton, and for vanishing zeta, as the type IIA D0-brane (for vanishing M-theory cicle radius). We find that the eta-function Seeley de-Witt coefficients vanish, and we get a perfect match between the zeta-function Seeley de-Witt coefficients (up to terms quadratic in zeta) between the Dirac-type operator and one of the two Laplace-type operators figuring in the superpotential. This is an extremely strong signature of residual supersymmetry for the nonperturbative configurations in M-theory considered in this work.

Laplace operators related to self-similar measures on [formula omitted

Given a bounded open subset Ω of R d ( d ⩾ 1 ) and a positive finite Borel measure μ supported on Ω ¯ with μ ( Ω ) > 0 , we study a Laplace-type operator Δ μ that extends the classical Laplacian. We show that the properties of this operator depend on the multifractal structure of the measure, especially on its lower L ∞ -dimension dim ̲ ∞ ( μ ) . We give a sufficient condition for which the Sobolev space H 0 1 ( Ω ) is compactly embedded in L 2 ( Ω , μ ) , which leads to the existence of an orthonormal basis of L 2 ( Ω , μ ) consisting of eigenfunctions of Δ μ . We also give a sufficient condition under which the Green's operator associated with μ exists, and is the inverse of − Δ μ . In both cases, the condition dim ̲ ∞ ( μ ) > d − 2 plays a crucial rôle. By making use of the multifractal L q -spectrum of the measure, we investigate the condition dim ̲ ∞ ( μ ) > d − 2 for self-similar measures defined by iterated function systems satisfying or not satisfying the open set condition.

Heat-kernel expansion on noncompact domains and a generalized zeta-function regularization procedure

Heat-kernel expansion and zeta function regularization are discussed for Laplace-type operators with discrete spectrum in noncompact domains. Since a general theory is lacking, the heat-kernel expansion is investigated by means of several examples. It is pointed out that for a class of exponential (analytic) interactions, generically the noncompactness of the domain gives rise to logarithmic terms in the heat-kernel expansion. Then, a meromorphic continuation of the associated zeta function is investigated. A simple model is considered, for which the analytic continuation of the zeta function is not regular at the origin, displaying a pole of higher order. For a physically meaningful evaluation of the related functional determinant, a generalized zeta function regularization procedure is proposed.

The ambient obstruction tensor and the conformal deformation complex

We construct here a conformally invariant differential operator on algebraic Weyl tensors that gives special curved analogues of certain operators related to the deformation complex and that, upon application to the Weyl curvature, yields the (Fefferman?Graham) ambient obstruction tensor. This new definition of the obstruction tensor leads to simple direct proofs that the obstruction tensor is divergence-free and vanishes identically for conformally Einstein metrics. Our main constructions are based on the ambient metric of Fefferman?Graham and its relation to the conformal tractor connection. We prove that the obstruction tensor is an obstruction to finding an ambient metric with curvature harmonic for a certain (ambient) form Laplacian. This leads to a new ambient formula for the obstruction in terms of a power of this form Laplacian acting on the ambient curvature. This result leads us to construct Laplacian-type operators that generalise the conformal Laplacians of Graham?Jenne?Mason?Sparling. We give an algorithm for calculating explicit formulae for these operators, and this is applied to give formulae for the obstruction tensor in dimensions 6 and 8. As background to these issues, we give an explicit construction of the deformation complex in dimensions n = 4, construct two related (detour) complexes, and establish essential properties of the operators in these.

High-energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators, and therefore the system remains noncommutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically.

Zeta regularized determinants for conic manifolds

We study (relative) zeta regularized determinants of Laplace type operators on compact conic manifolds. We establish gluing formulae for relative zeta regularized determinants. For arbitrary self-adjoint extensions of the Laplace–Beltrami operator, we express the relative ζ-determinants for these as a ratio of the determinants of certain finite matrices. For the self-adjoint extensions corresponding to Dirichlet and Neumann conditions, the formula is particularly simple and elegant.

Laplacian Operators and Q-curvature on Conformally Einstein Manifolds

A new definition of canonical conformal differential operators P k (k = 1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction these factor into a power of a fundamental Laplacian associated to Einstein metrics. There are natural conformal Laplacian operators on density bundles due to Graham–Jenne–Mason–Sparling (GJMS). It is shown that on conformally Einstein manifolds these agree with the P k operators and hence on Einstein manifolds the GJMS operators factor into a product of second-order Laplacian type operators. In even dimension n the GJMS operators are defined only for 1 ≤ k ≤ n/2 and so, on conformally Einstein manifolds, the P k give an extension of this family of operators to operators of all even orders. For n even and k > n/2 the operators P k are each given by a natural formula in terms of an Einstein metric but they are not natural conformally invariant operators in the usual sense. They are shown to be nevertheless canonical objects on conformally Einstein structures. There are generalisations of these results to operators between weighted tractor bundles. It is shown that on Einstein manifolds the Branson Q-curvature is constant and an explicit formula for the constant is given in terms of the scalar curvature. As part of development, conformally invariant tractor equations equivalent to the conformal Killing equation are presented.

Regularized determinants of Laplace-type operators, analytic surgery, and relative determinants

Let M be a compact Riemannian manifold in which Y is an embedded hypersurface separating M into two parts. Assume that the metric is a product on a tubular neighborhood N of Y. Let Δ be a Laplace-type operator on M adapted to the product structure on N. Under certain additional assumptions on Δ, we establish an asymptotic expansion for the logarithm of the regularized determinant detΔ of Δ if the tubular neighborhood N is stretched to a cylinder of infinite length. We use the asymptotic expansions to derive adiabatic splitting formulas for regularized determinants

Zeta functions of Dirac and Laplace-type operators over finite cylinders

In this paper, a complete description of the zeta functions and corresponding zeta determinants for Dirac and Laplace-type operators over finite cylinders using the contour integration method, for example described in [K. Kirsten, Spectral Functions in Mathematics and Physics, Chapman & Hall/CRC Press, Boca Raton, 2001] is given. Different boundary conditions, local and non-local ones, are considered. The method is shown to be very powerful in that it is easily adapted to each situation and in that answers are very elegantly obtained.

Relative Superlinearity Implies Infinitely Many Solutions to Sturm–Liouville BVPs with Laplacian Type Operators

By the use of continuation theorems and the duality principle it’s proved that for both homogeneous and nonhomogeneous Sturm–Liouville boundary value problems with Laplacian type operators, relative superlinearity implies the existence of infinitely many solutions under a few weakly added conditions.

DIRAC OPERATOR IN MATRIX GEOMETRY

We review the construction of the Dirac operator and its properties in Riemannian geometry, and show how the asymptotic expansion of the trace of the heat kernel determines the spectral invariants of the Dirac operator and its index. We also point out that the Einstein–Hilbert functional can be obtained as a linear combination of the first two spectral invariants of the Dirac operator. Next, we report on our previous attempts to generalize the notion of the Dirac operator to the case of Matrix Geometry, where, instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays a role of a "non-commutative" metric. We construct invariant first-order and second-order self-adjoint elliptic partial differential operators, which can be called "non-commutative" Dirac operators and non-commutative Laplace operators. We construct the corresponding heat kernel for the non-commutative Laplace type operator and compute its first two spectral invariants. A linear combination of these two spectral invariants gives a functional that can be considered as a non-commutative generalization of the Einstein–Hilbert action.

Spectral asymptotics of Euclidean quantum gravity with diff-invariant boundary conditions

A general method is known to exist for studying Abelian and non-Abelian gauge theories, as well as Euclidean quantum gravity, at 1-loop level on manifolds with boundary. In the latter case, boundary conditions on metric perturbations h can be chosen to be completely invariant under infinitesimal diffeomorphisms, to preserve the invariance group of the theory and BRST symmetry. In the de Donder gauge, however, the resulting boundary-value problem for the Laplace-type operator acting on h is known to be self-adjoint but not strongly elliptic. The latter is a technical condition ensuring that a unique smooth solution of the boundary-value problem exists, which implies, in turn, that the global heat-kernel asymptotics yielding 1-loop divergences and 1-loop effective action actually exists. The present paper shows that, on the Euclidean 4-ball, only the scalar part of perturbative modes for quantum gravity is affected by the lack of strong ellipticity. Further evidence for lack of strong ellipticity, from an analytic point of view, is therefore obtained. Interestingly, three sectors of the scalar-perturbation problem remain elliptic, while lack of strong ellipticity is ‘confined’ to the remaining fourth sector. The integral representation of the resulting ζ-function asymptotics on the Euclidean 4-ball is also obtained; this remains regular at the origin by virtue of a spectral identity here obtained for the first time.

Adiabatic decomposition of the [formula omitted]-determinant and Dirichlet to Neumann operator

We discuss the adiabatic decomposition formula of the ζ -determinant of a Laplace type operator on a closed manifold. We also analyze the adiabatic behavior of the ζ -determinant of a Dirichlet to Neumann operator. This analysis makes it possible to compare the adiabatic decomposition formula with the Mayer–Vietoris type formula for the ζ -determinant proved by Burghelea et al. As a byproduct of this comparison, we obtain the exact value of the local constant which appears in their formula for the case of Dirichlet boundary condition.

Zeta functions in brane world cosmology

We present a calculation of the zeta function and of the functional determinant for a Laplace-type differential operator, corresponding to a scalar field in a higher-dimensional de Sitter brane background, which consists of a higher-dimensional anti de Sitter bulk spacetime bounded by a de Sitter section, representing a brane. Contrary to the existing examples, which all make use of conformal transformations, we evaluate the zeta function working directly with the higher-dimensional wave operator. We also consider a generic mass term and coupling to curvature, generalizing previous results. The massless, conformally coupled case is obtained as a limit of the general result and compared with known calculations. In the limit of large anti de Sitter radius, the zeta determinant for the ball is recovered in perfect agreement with known expressions, providing an interesting check of our result and an alternative way of obtaining the ball determinant.

Decomposition of the {$\zeta$}-determinant for the Laplacian on manifolds with cylindrical end

In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the decomposition problem for the (regularized) �-determinant of the Laplacian on a manifold with cylin- drical end into the �-determinants of the Laplacians with Dirichlet con- ditions on the manifold with boundary and on the half infinite cylinder. We also compute all the contributions to this formula explicitly. We investigate the 'Mayer-Vietoris' or 'cut and paste' decomposition for- mula of the �-determinant for a Laplacian on a manifold with cylindrical end into the �-determinants of the Laplacians with Dirichlet conditions on the manifold with boundary and on the half infinite cylinder. We also introduce a new method to attack such surgery problems by comparing the problem to a corresponding model problem. This approach works for compact manifolds as well as manifolds with cylindrical ends and will be used to solve related decomposition problems for the spectral invariants of Dirac type operators in (12), (13). We remark that the noncompactness of the underlying manifold introduces many new facets and obstacles not found in the compact case, as we will explain later. We begin with a brief account of zeta determinants. The �-determinant of a Laplace type operator was pioneered in the seminal paper (21) by Ray and Singer. They were seeking an analytic version of the so- called Reidemeister torsion, a combinatorial-topological invariant introduced by Reidemeister (22) and Franz (6). They conjectured that their analytic in- variant was the same as the Reidemeister torsion. Later, this conjecture was proved independently by Cheeger (4) and Muller (17). The �-determinants have also been of great use in quantum field theory where they are being used to develop rigorous models for Feynman path integral techniques (10). Because of their use in differential topology and quantum field theory, much work has been done on understanding the nature of�-determinants, especially their behavior under 'cutting and pasting' of manifolds. This was initiated

Hessians of spectral zeta functions

Let M be a compact manifold without boundary. Associated to a metric g on M there are various Laplace operators, for example the de Rham Laplacian on forms and the conformal Laplacian on functions. For a general Laplace type operator we consider its spectral zeta function Z(s). For a fixed value of s we calculate the hessian of Z(s) with respect to the metric and show that it is given by a pseudodifferential operator T(s)=U(s)+V(s) where U(s) is polyhomogeneous of degree n-2s and V(s) is polyhomogeneous of degree 2. The operators U(s) and V(s) are meromorphic in s. The symbol expansion of U(s) is computable from the symbol of the Laplacian and we give an explicit formula for the principal symbol. Our analysis extends to describe the hessian of the kth order derivative of Z(s) with respect to s, in particular to the hessian of Z'(0).

Noncommutative Heat Kernel

We consider a natural generalisation of the Laplace type operators for the case of noncommutative (Groenewold—Moyal star) product. We demonstrate existence of a power law asymptotic expansion for the heat trace of such operators on Tn. First four coefficients of this expansion are calculated explicitly. We also find an analog of the UV/IR mixing phenomenon when analysing the localised heat kernel